A lot depends on the energy of the gamma-ray beam as a whole and the energy of the gammas themselves. The more energetic they are, the deeper they penetrate - which means they might come out on the other side of the target without having deposited much energy and doing damage.
For low-energy 100 MeV gammas the attenuation length (the distance where 60% of them gets absorbed) in water is about 4 cm, while for 2000 MeV gammas it is more than 10 cm. Lead is about a factor of 10 shorter. So if you fired your 2000 MeV beam at a person a fair bit of energy would just pass through, while the 100 MeV beam would all deposit it in the first few centimetres.
The absorbed energy leads to electrons and x-ray photons bouncing away in all directions, further interacting with the target. In normal radiation physics this is where things would be complicated because we would try to estimate the biological effects, many of which are long-term. But if you are just pouring on a lot of energy eventual cancer risk is hardly the main problem of the target. Basically you will heat a volume of tissue (or whatever else you are hitting) with the energy of the death ray.
This is where the overall power of the beam matters: if it is just a few Watt it will just gently heat the target. If it is in the kilowatt range it will increase temperature of heavy targets by hundreds of degrees (and less for watery targets since the heating gets spread out over a much larger volume), if it is in the megawatt range we will presumably see a steam explosion if you hit somebody (the heat of vaporisation for water is 2257 J/g, so if the affected volume is about a litre you will supply that energy in 2.3 seconds with a 1 MW beam).
In the end, what is doing the damage is the total energy supply rather than it being gamma rays, although having the energy absorbed in the volume rather than on the surface might have important effects on the type of damage.